**Uncertainty relations: An operational approach to the error-disturbance tradeoff**

Joseph M. Renes1_{, Volkher B. Scholz}1,2

_{, and Stefan Huber}1,3

*1 _{Institute for Theoretical Physics, ETH Zürich, Switzerland}*

*2*

_{Department of Physics, Ghent University, Belgium}*3 _{Department of Mathematics, Technische Universität München, Germany}*

Accepted in

### Quantum

July 10, 2017The notions of error and disturbance appearing in quantum uncertainty relations are often quantified
by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not
the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not
necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take
a different approach and define error and disturbance in an operational manner. In particular, we
formu-late both in terms of the probability that one can successfully distinguish the actual measurement device
from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely
on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We
then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance
tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and
mo-mentum. Our relations may be directly applied in information processing settings, for example to infer that
devices which can faithfully transmit information regarding one observable do not leak any information
about conjugate observables to the environment. We also show that Englert’s wave-particle duality relation
[Phys. Rev. Lett.**77, 2154 (1996)**_{]}can be viewed as an error-disturbance uncertainty relation.

**1** **Introduction**

It is no overstatement to say that the uncertainty principle is a cornerstone of our understanding of
quan-tum mechanics, clearly marking the departure of quanquan-tum physics from the world of classical physics.
Heisen-berg’s original formulation in 1927 mentions two facets to the principle. The first restricts the joint
measur-ability of observables, stating that noncommuting observables such as position and momentum can only be
simultaneously determined with a characteristic amount of indeterminacy_{[}1, p. 172_{]}(see_{[}2, p. 62_{]}for an
English translation). The second describes an error-disturbance tradeoff, noting that the more precise a
mea-surement of one observable is made, the greater the disturbance to noncommuting observables[1, p. 175]
([2, p. 64]). The two are of course closely related, and Heisenberg argues for the former on the basis of the
latter. Neither version can be taken merely as a limitation on measurement of otherwise well-defined values
of position and momentum, but rather as questioning the sense in which values of two noncommuting
ob-servables can even be said to simultaneously exist. Unlike classical mechanics, in the framework of quantum
mechanics we cannot necessarily regard unmeasured quantities as physically meaningful.

More formal statements were constructed only much later, due to the lack of a precise mathematical
description of the measurement process in quantum mechanics. Here we must be careful to draw a distinction
between statements addressing Heisenberg’s original notions of uncertainty from those, like the standard
Kennard-Robertson uncertainty relation[3,4], which address the impossibility of finding a quantum state
with well-defined values for noncommuting observables. Entropic uncertainty relations[5,6]are also an
example of this class; see[7]for a review. Joint measurability has a longer history, going back at least to
the seminal work of Arthurs and Kelly_{[}8_{]}and continuing in_{[}9–27_{]}. Quantitative error-disturbance relations
have only been formulated relatively recently, going back at least to Braginsky and Khalili_{[}28, Chap. 5_{]}and
continuing in_{[}20,29–35_{]}.

Beyond technical difficulties in formulating uncertainty relations, there is a perhaps more difficult con-ceptual hurdle in that the intended consequences of the uncertainty principle seem to preclude their own straightforward formalization. To find a relation between, say, the error of a position measurement and its disturbance to momentum in a given experimental setup like the gamma ray microscope would seem to re-quire comparing the actual values of position and momentum with their supposed ideal values. However, according to the uncertainty principle itself, we should be wary of simultaneously ascribing well-defined val-ues to the actual and ideal position and momentum since they do not correspond to commuting observables. Thus, it is not immediately clear how to formulate either meaningful measures of error and disturbance, for instance as mean-square deviations between real and ideal values, or a meaningful relation between them.1 This question is the subject of much ongoing debate[25,30,36–39].

1_{Uncertainty relations like the Kennard-Robertson bound or entropic relations do not face this issue as they do not attempt to compare}
actual and ideal values of the observables.

Without drawing any conclusions as to the ultimate success or failure of this program, in this paper we
pro-pose a completely different approach which we hope sheds new light on these conceptual difficulties. Here,
we define error and disturbance in an operational manner and ask for uncertainty relations that are
state-ments about the properties of measurement devices, not of fixed experimental setups or of physical quantities
themselves. More specifically, we define error and disturbance in terms of the*distinguishing probability*, the
probability that the actual behavior of the measurement apparatus can be distinguished from the relevant ideal
behavior in any single experiment whatsoever. To characterize measurement error, for example, we imagine
a black box containing either the actual device or the ideal device. By controlling the input and observing the
output we can make an informed guess as to which is the case. We then attribute a large measurement error
to the measurement apparatus if it is easy to tell the difference, so that there is a high probability of correctly
guessing, and a low error if not; of course we pick the optimal input states and output measurements for this
purpose. In this way we do not need to attribute a particular ideal value of the observable to be measured,
we do not need to compare actual and ideal values themselves (nor do we necessarily even care what the
possible values are), and instead we focus squarely on the properties of the device itself. Intuitively, we might
expect that calibration provides the strictest test, i.e. inputting states with a known value of the observable
in question. But in fact this is not the case, as entanglement at the input can increase the distinguishability
of two measurements. The merit of this approach is that the notion of distinguishability itself does not rely
on any concepts or formalism of quantum theory, which helps avoid conceptual difficulties in formalizing the
uncertainty principle.

Defining the disturbance an apparatus causes to an observable is more delicate, as an observable itself does not have a directly operational meaning (as opposed to the measurement of an observable). But we can consider the disturbance made either to an ideal measurement of the observable or to ideal preparation of states with well-defined values of the observable. In all cases, the error and disturbance measures we consider are directly linked to a well-studied norm on quantum channels known as the completely bounded norm or diamond norm. We can then ask for bounds on the error and disturbance quantities for two given observables that every measurement apparatus must satisfy. In particular, we are interested in bounds de-pending only on the chosen observables and not the particular device. Any such relation is a statement about measurement devices themselves and is not specific to the particular experimental setup in which they are used. Nor are such relations statements about the values or behavior of physical quantities themselves. In this sense, we seek statements of the uncertainty principle akin to Kelvin’s form of the second law of thermo-dynamics as a constraint on thermal machines, and not like Clausius’s or Planck’s form involving the behavior of physical quantities (heat and entropy, respectively). By appealing to a fundamental constraint on quan-tum dynamics, the continuity (in the completely bounded norm) of the Stinespring dilation[40, 41], we find error-disturbance uncertainty relations for arbitrary observables in finite dimensions, as well as for po-sition and momentum. Furthermore, we show how the relation for measurement error and measurement disturbance can be transformed into a joint-measurability uncertainty relation. Interestingly, we also find that Englert’s wave-particle duality relation[42]can be viewed as an error-disturbance relation.

The case of position and momentum illustrates the stark difference between the kind of uncertainty
state-ments we can make in our approach with one based on the notion of comparing real and ideal values. Take
the notion of joint measurability, where we would like to formalize the notion that no device can accurately
measure both position and momentum. In the latter approach one would first try to quantify the amount
of position or momentum error made by a device as the discrepancy to the true value, and then show that
they cannot both be small. The errors would be in units of position or momentum, respectively, and the
hoped-for uncertainty relation would pertain to these values. Here, in contrast, we focus on the performance
of the actual device relative to fixed ideal devices, in this case idealized separate measurements of position or
momentum. Importantly, we need not think of the ideal measurement as having infinite precision. Instead,
we can pick any desired precision and ask if the behavior of the actual device is essentially the same as this
precision-limited ideal. Now the position and momentum errors do not have units of these quantities (they
are unitless and always lie between zero and one), but instead*depend on the desired precision*. Our
uncer-tainty relation then implies that both errors cannot be small if we demand high precision in both position
and momentum. In particular, when the product of the scales of the two precisions is small compared to
Planck’s constant, then the errors will be bounded away from zero (see Theorem3for a precise statement).
It is certainly easier to have a small error in this sense when the demanded precision is low, and this accords
nicely with the fact that sufficiently-inaccurate joint measurement is possible. Indeed, we find no bound on
the errors for low precision.

in establishing simple proofs of the security of quantum key distribution[6,7,43–45]. Here we show that the error-disturbance relation implies that quantum channels which can faithfully transmit information regarding one observable do not leak any information whatsoever about conjugate observables to the environment. This statement cannot be derived from entropic relations, as it holds for all channel inputs. It can be used to construct leakage-resilient classical computers from fault-tolerant quantum computers[46], for instance.

The remainder of the paper is structured as follows. In the next section we give the mathematical back-ground necessary to state our results, and describe how the general notion of distinguishability is related to the completely bounded norm (cb norm) in this setting. In Section3we define our error and disturbance measures precisely. Section4presents the error-disturbance tradeoff relations for finite dimensions, and de-tails how joint measurability relations can be obtained from them. Section5considers the error-disturbance tradeoff relations for position and momentum. Two applications of the tradeoffs are given in Section6: a for-mal statement of the information disturbance tradeoff for information about noncommuting observables and the connection between error-disturbance tradeoffs and Englert’s wave-particle duality relations. In Section7 we compare our results to previous approaches in more detail, and finally we finish with open questions in Section8.

**2** **Mathematical setup**
**2.1** **Distinguishability**

The notion of the distinguishing probability is independent of the mathematical framework needed to describe
quantum systems, so we give it first. Consider an apparatus_{E} which in some way transforms an input*A*into
an output*B*. To describe how different_{E} is from another such apparatus_{E}0_{, we can imagine the following}
scenario. Suppose that we randomly place either_{E} or_{E}0 _{into a black box such that we no longer have any}
access to the inner workings of the device, only its inputs and outputs. Now our task is to guess which device
is actually in the box by performing a single experiment, feeding in any desired input and observing the output
in any manner of our choosing. In particular, the inputs and measurements can and should depend on_{E}and
E0. The probability of making a correct guess, call it*p*dist(E,E0), ranges from 12to 1, since we can always just
make a random guess without doing any experiment on the box at all. Therefore it is more convenient to
work with the distinguishability measure

*δ*(E,E0):=2*p*dist(E,E0)−1 , (1)

which ranges from zero (completely indistinguishable) to one (completely distinguishable). Later on we will
show this quantity takes a specific mathematical form in quantum mechanics. But note that the definition
implies that the distinguishability is monotonic under concatenation with a channelF to bothEandE0, since
this just restricts the possible tests. That is, both*δ*(EF,E0F)≤*δ*(E,E0)and*δ*(FE,FE0)≤*δ*(E,E0)hold for
all channels_{F} whose inputs and outputs are such that the channel concatenation is sensible. Here and in
the remainder of the paper, we denote concatenation of channels by juxtaposition, while juxtaposition of
operators denotes multiplication as usual.

**2.2** **Systems, algebras, channels, and measurements**

In the finite-dimensional case we will be interested in two arbitrary nondegenerate observables denoted*X*
and*Z*. Only the eigenvectors of the observables will be relevant, call them|*ϕx*〉and|*θz*〉, respectively. In

infinite dimensions we will confine our analysis to position*Q*and momentum*P*, takingħ*h*=1. The analog of
*Q*and*P*in finite dimensions are canonically conjugate observables*X*and*Z*for which|*ϕx*〉= p1* _{d}*P

*zωxz*|

*θz*〉,

where*d*is the dimension and*ω*is a primitive*d*th root of unity.

It will be more convenient for our purposes to adopt the algebraic framework and use the Heisenberg
picture, though we shall occasionally employ the Schrödinger picture. In the Heisenberg picture we describe
systems chiefly by the algebra of observables on them and describe transformations of systems by quantum
channels, completely positive and unital maps from the algebra of observables of the output to the observables
of the input[10,47–50]. This allows us to treat classical and quantum systems on an equal footing within
the same framework. When the input or output system is quantum mechanical, the observables are the
bounded operators_{B}_{(}_{H}_{)}from the Hilbert space_{H}associated with the system to itself. Classical systems,
such as the results of measurement or inputs to a state preparation device, take values in a set, call itY.

For arbitrary input and output algebras_{A}*A*andA*B*, quantum channels are precisely those mapsE which

are unital,_{E}_{(}1*B*) =1*A*, and completely positive, meaning that not only doesE map positive elements ofA*B*

to positive elements of_{A}*A*, it also maps positive elements ofA*B*⊗B(C*n*)to positive elements ofA*A*⊗B(C*n*)

for all integer*n*. This requirement is necessary to ensure that channels act properly on entangled systems.

*A* _{E} *B*

Y

Figure 1: A general quantum apparatus_{E}. The apparatus measures a quantum system*A*giving the
outputY. In so doing,_{E} also transforms the input*A*into the output system*B*. Here the wavy lines
denote quantum systems, the dashed lines classical systems. Formally, the apparatus is described by
a quantum instrument.

A general measurement apparatus has both classical and quantum outputs, corresponding to the
mea-surement result and the post-meamea-surement quantum system. Channels describing such devices are called
*quantum instruments*; we will call the channel describing just the measurement outcome a*measurement*. In
finite dimensions any measurement can be seen as part of a quantum instrument, but not so for idealized
position or momentum measurements, as shown in Theorem 3.3 of[10](see page 57). Technically, we may
anticipate the result since the post-measurement state of such a device would presumably be a delta function
located at the value of the measurement, which is not an element of*L*2_{(}_{Q}_{)}_{. This need not bother us, though,}
since it is not operationally meaningful to consider a position measurement instrument of infinite precision.
And indeed there is no mathematical obstacle to describing finite-precision position measurement by
quan-tum instruments, as shown in Theorem 6.1 (page 67 of_{[}10_{]}). For any bounded function*α*∈*L*2_{(}_{Q}_{)}_{we can}
define the instrument_{E}* _{α}*:

*L*∞

_{(}

_{Q}

_{)}

_{⊗}

_{B}

_{(}

_{H}

_{)}

_{→}

_{B}

_{(}

_{H}

_{)}

_{by}

E*α(f* ⊗*a*_{) =}
Z

d*q f*_{(}*q*_{)}*A*∗

*q*;*αaAq*;*α*, (2)

where*Aq*;*αψ*(*q*0) =*α*(*q*−*q*0)*ψ*(*q*0)for all*ψ* ∈ *L*2(Q). The classical output of the instrument is essentially
the ideal value convolved with the function*α. Thus, setting the width of* *α*sets the precision limit of the
instrument.

**2.3** **Distinguishability as a channel norm**

The distinguishability measure is actually a norm on quantum channels, equal (apart from a factor of one
half) to the so-called norm of complete boundedness, the cb norm_{[}51–53_{]}. The cb norm is defined as an
extension of the operator norm, similar to the extension of positivity above, as

k*T*_{k}_{cb}:_{=}sup

*n*∈N

k1*n*⊗*T*k∞, (3)

where_{k}*T*_{k}_{∞}is the operator norm. Then

*δ*(E1,E2) =12kE1−E2kcb. (4)

In the Schrödinger picture we instead extend the trace norm_{k·k}1, and the result is usually called the diamond
norm[51,53]. In either case, the extension serves to account for entangled inputs in the experiment to test
whether_{E1} or_{E2} is the actual channel. In fact, entanglement is helpful even when the channels describe
projective measurements, as shown by an example given in AppendixA. This expression for the cb or diamond
norm is not closed-form, as it requires an optimization. However, in finite dimensions the cb norm can be cast
as a convex optimization, specifically as a semidefinite program[54,55], which makes numerical computation
tractable. Further details are given in AppendixB.

**2.4** **The Stinespring representation and its continuity**

According to the Stinespring representation theorem[52,56], any channel_{E} mapping an algebra_{A}to_{B}_{(}_{H}_{)}
can be expressed in terms of an isometry*V* :_{H}_{→}_{K}to some Hilbert space_{K}and a representation*π*ofAin
B(K)such that, for all*a*∈A,

The isometry in the Stinespring representation is usually called the*dilation*of the channel, and_{K}the dilation
space. In finite-dimensional settings, calling the input*A*and the output*B*, one usually considers maps taking
A=B(H*B*)toB(H*A*). Then one can chooseK=H*B*⊗H*E*, whereH*E*is a suitably large Hilbert space associated

to the “environment” of the transformation (_{H}*E* can always be chosen to have dimension dim(H*A*)dim(H*B*)).

The representation*π*is just*π*(*a*) =*a*⊗1*E*. Using the isometry*V*, we can also construct a channel fromB(H*E*)

to_{B}_{(}_{H}*A*)in the same manner; this is known as the complementE*]*ofE.

The advantage of the general form of the Stinespring representation is that we can easily describe
mea-surements, possibly continuous-valued, as well. For the case of finite outcomes, consider the ideal projective
measurementQ*X* of the observable*X*. Choosing a basis{|*bx*〉}of*L*2(X)and defining*π*(*δx*) =|*bx*〉〈*bx*|for*δx*

the function taking the value 1 at*x*and zero elsewhere, the canonical dilation isometry*W _{X}*:H→

*L*2

_{(}

_{X}

_{)}

_{⊗}

_{H}is given by

*WX*=

X

*x*

|*bx*〉 ⊗ |*ϕx*〉〈*ϕx*|. (6)

Note that this isometry defines a quantum instrument, since it can describe both the measurement outcome
and the post-measurement quantum system. If we want to describe just the measurement result, we could
simply use*WX*=P*x*|*bx*〉 〈*ϕx*|with the same*π. More generally, a POVM with elementsΛx* has the isometry

*WX* =P*x*|*bx*〉 ⊗p*Λx*.

For finite-precision measurements of position or momentum, the form of the quantum instrument in (2)
immediately gives a Stinespring dilation*W _{Q}*:

_{H}→KwithK=

*L*2(Q

_{)}⊗

_{H}whose action is defined by

(*WQψ*)(*q*,*q*0) =*α*(*q*−*q*0)*ψ*(*q*0), (7)

and where*π*is just pointwise multiplication on the*L*∞(Q)factor, i.e. for *f* ∈*L*∞_{(}_{Q}_{)}_{, and}_{a}_{∈}_{B}_{(}_{H}_{)}_{,}_{[}_{π}_{(}_{f}_{⊗}
*a*_{)(}*ξ*⊗*ψ*)](*q*,*q*0) =*f*(*q*)*ξ*(*q*)·(*aψ*)(*q*0)for all*ξ*∈*L*2_{(}_{Q}_{)}_{and}_{ψ}_{∈}_{H}_{.}

A slight change to the isometry in (6) gives the dilation of the device which prepares the state _{|}*ϕx*〉

for classical input *x*. Formally the device is described by the map _{P} :_{B}_{(}_{H}_{)} _{→} *L*2_{(}_{X}_{)}_{for which}_{P}_{(}_{Λ}_{) =}
P

*x*|*bx*〉〈*bx*| 〈*ϕx*|*Λ*|*ϕx*〉. Now consider*WX*0 :*L*2(X)→H⊗*L*2(X)given by

*W*0

*X*=

X

*x* |

*ϕx*〉 ⊗ |*bx*〉〈*bx*|. (8)

Choosing*π*_{(}*Λ*_{) =}*Λ*⊗1X, we haveP(*Λ*) =*WX*0∗*π*(*Λ*)*WX*0.

The Stinespring representation is not unique[41]. Given two representations_{(}*π*1,*V*1,K1)and(*π*2,*V*2,K2)
of the same channel_{E}, there exists a partial isometry*U* :_{K1}→K2 such that*UV*1 = *V*2, *U*∗*V*2 = *V*1, and
*Uπ*1(*a*) =*π*2(*a*)*U* for all*a*∈A. For the representations*π*as usually employed for the finite-dimensional
case, this last condition implies that*U*is a partial isometry from one environment to the other, for*U*_{(}*a*_{⊗}1*E*) =

(*a*⊗1*E*0)*U* can only hold for all*a*if*U* acts trivially on*B*. For channels describing measurements, finite or
continuous, the last condition implies that any such*U* is a conditional partial isometry, dependent on the
outcome of the measurement result. Thus, for any set of isometries*U _{x}*:

_{H}

_{S}_{→}

_{H}

*,P*

_{R}

_{x}_{|}

*b*

_{x}_{〉 ⊗}

*U*

_{x}_{|}

*ϕx*〉〈

*ϕx*|

*Ux*∗

is a valid dilation of_{Q}*X*, just as is*WX* in (6). Similarly,(*WQ*0*ψ*)(*q*,*q*0) =*α*(*q*−*q*0)[*Uqψ*](*q*0)is a valid dilation

of_{E}*α*in (2).

The main technical ingredient required for our results is the continuity of the Stinespring representation
in the cb norm[40,41]. That is, channels which are nearly indistinguishable have Stinespring dilations which
are close and vice versa. For completely positive and unital maps_{E1}and_{E2},[40,41]show that

1

2kE1−E2kcb≤ inf
*πi*,*Vi*k

*V*_{1}_{−}*V*_{2}_{k}_{∞}_{≤}Æ_{k}_{E1}_{−}_{E2}_{k}_{cb}, (9)

where the infimum is taken over all Stinespring representations_{(}*πi*,*Vi*,K*i*)ofE*i*.

**2.5** **Sequential and joint measurements**

Using the Stinespring representation we can easily show that, in principle, any joint measurement can always be decomposed into sequential measurement.

**Lemma 1.** *Suppose that*_{E}:*L*∞_{(}X_{)}_{⊗}*L*∞_{(}Z_{)}_{→}_{B}_{(}_{H}_{)}*is a channel describing a joint measurement. Then*

*Proof.* Define_{M}0 _{:}* _{L}*∞

_{(}

_{X}

_{)}

_{→}

_{B}

_{(}

_{H}

_{)}

_{to be just the}

_{X}

_{output of}

_{E}

_{, i.e.}

_{M}0

_{(}

_{f}_{) =}

_{E}

_{(}

_{f}_{⊗}

_{1}

_{)}

_{. Now suppose that}

*V*:

_{H}

_{→}

*L*2

(X_{)}_{⊗}*L*2_{(}Z_{)}_{⊗}_{H}00is a Stinespring representation of_{E} and*VX*:H→*L*2(X)⊗H0is a representation

of_{M}0_{, both with the standard representation}_{π}_{of} * _{L}*∞

_{into}

*2*

_{L}_{. By construction,}

_{V}_{is also a dilation of}M0, and therefore there exists a partial isometry

*UX*such that

*V*=

*UXVX*. More specifically, conditional on the

valueX _{=} *x*, each *Ux* sendsH0 to *L*2(Z)⊗H00. Thus, settingA(*f* ⊗*a*) = *VX*∗(*π*(*f*)⊗*a*)*VX* andM*x*(*f*) =

*U*∗

*x*(*π*(*f*)⊗1)*Ux*, we haveE=AM.

**3** **Definitions of error and disturbance**
**3.1** **Measurement error**

To characterize the error*"X* an apparatusE makes relative to an ideal measurementQ*X* of an observable*X*,

we can simply use the distinguishability of the two channels, taking only the classical output of_{E}. Suppose
that the apparatus is described by the channel _{E} :_{B}_{(}_{H}*B*)⊗ *L*∞(X) →B(H*A*)and the ideal measurement

by the channel _{Q}*X* : *L*∞(X)→ B(H*A*). To ignore the output system *B*, we make use of the partial trace

map_{T}*B* :*L*∞(X)→B(H*B*)⊗*L*∞(X)given byT*B*(*f*) =1*B*⊗ *f*. Then a sensible notion of error is given by

*"X*(E) =*δ*(Q*X*,ET*B*). If it is easy to tell the ideal measurement apart from the actual device, then the error is

large; if it is difficult, then the error is small.

As a general definition, though, this quantity is deficient to two respects. First, we could imagine an
apparatus which performs an ideal_{Q}*X* measurement, but simply mislabels the outputs. This leads to*"X*(E) =

1, even though the ideal measurement is actually performed. Second, we might wish to consider the case that
the classical output set of the apparatus is not equal toXitself. For instance, perhapsE delivers much more
output than is expected from_{Q}* _{X}*. In this case we also formally have

*"X*(E) =1, since we can just examine the

output to distinguish the two devices.

We can remedy both of these issues by describing the apparatus by the channel_{E} :_{B}_{(}_{H}_{B}_{)}_{⊗}*L*∞_{(}_{Y}_{)}_{→}
B(H*A*)and just including a further classical postprocessing operationR : *L*∞(X)→ *L*∞(Y)in the

distin-guishability step. Since we are free to choose the best such map, we define

*"X*(E):=inf

R *δ*(Q*X*,ERT*B*). (10)

The setup of the definition is depicted in Figure2.

*A* _{E} _{R} _{X} ≈*"X* *A* Q*X* X

*B*

Y

Figure 2: Measurement error. The error made by the apparatusE in measuring*X* is defined by how
distinguishable the actual device is from the ideal measurement_{Q}* _{X}* in any experiment whatsoever,
after suitably processing the classical outputYof

_{E}with the map

_{R}. To enable a fair comparison, we

ignore the quantum output of the apparatus, indicated in the diagram by graying out*B*. If the actual
and ideal devices are difficult to tell apart, the error is small.

**3.2** **Measurement disturbance**

Defining the disturbance an apparatus_{E}causes to an observable, say*Z*, is more delicate, as an observable itself
does not have a directly operational meaning. But there are two straightforward ways to proceed: we can
either associate the observable with measurement or with state preparation. In the former, we compare how
well we can mimic the ideal measurementQ*Z*of the observable after employing the apparatusE, quantifying

this using measurement error as before. Additionally, we should allow the use of recovery operations in
which we attempt to “restore” the input state as well as possible, possibly conditional on the output of the
measurement. Formally, let _{Q}* _{Z}* :

*L*∞

_{(}

_{Z}

_{)}

_{→}

_{B}

_{(}

_{H}

*A*) be the ideal *Z* measurement and R be a recovery

map_{R}:_{B}_{(}_{H}_{A}_{)}_{→}_{B}_{(}_{H}_{B}_{)}_{⊗}*L*∞_{(}_{X}_{)}_{which acts on the output of}_{E} _{conditional on the value of the classical}
outputX(which it then promptly forgets). As depicted in Figure3, the measurement disturbance is then the

measurement error after using the best recovery map:

*νZ*(E):=inf

*A* _{E} _{R} Q*Z* Z ≈*νZ* *A* Q*Z* Z

Y

Figure 3: Measurement disturbance. To define the disturbance imparted by an apparatus_{E} to the
measurement of an observable *Z*, consider performing the ideal_{Q}* _{Z}* measurement on the output

*B*of

_{E}. First, however, it may be advantageous to “correct” or “recover” the original input

*A*by some operation

_{R}. In general,

_{R}may depend on the outputXof

_{E}. The distinguishability between the

resulting combined operation and just performing_{Q}*Z*on the original input defines the measurement

disturbance.

**3.3** **Preparation disturbance**

For state preparation, consider a device with classical input and quantum output that prepares the eigenstates
of*Z*. We can model this by a channel_{P}*Z*, which in the Schrödinger picture produces|*θz*〉upon receiving the

input*z*. Now we compare the action of_{P}*Z* to the action ofP*Z* followed byE, again employing a recovery

operation. Formally, let _{P}*Z* : B(H*A*) → *L*∞(Z) be the ideal *Z* preparation device and consider recovery

operations_{R}of the form_{R}:_{B}_{(}_{H}*A*)→B(H*B*)⊗*L*∞(X). Then the preparation disturbance is defined as

*ηZ*(E):=inf_{R}*δ*(P*Z*,P*Z*ERTY). (12)

Z P*Z* E R *A* ≈*ηZ* Z P*Z* *A*

Y

Figure 4: Preparation disturbance. The ideal preparation device _{P}*Z* takes a classical inputZ and

creates the corresponding *Z* eigenstate. As with measurement disturbance, the preparation
distur-bance is related to the distinguishability of the ideal preparation device_{P}*Z* andP*Z* followed by the

apparatus_{E} in question and the best possible recovery operation_{R}.

All of the measures defined so far are “figures of merit”, in the sense that we compare the actual device to
the ideal, perfect functionality. In the case of state preparation we can also define a disturbance measure as a
“figure of demerit”, by comparing the actual functionality not to the best-case behavior but to the worst. To
this end, consider a state preparation device_{C}which just ignores the classical input and always prepares the
same fixed output state. These are constant (output) channels, and clearly_{E}disturbs the state preparation_{P}* _{Z}*
considerably if

_{P}

*Z*Ehas effectively a constant output. Based on this intuition, we can then make the following

formal definition:

b

*ηZ*(E):= *d*−1*d* −_{C}_{:const.}inf *δ*(C,P*Z*E). (13)

The disturbance is small according to this measure if it is easy to distinguish the action of_{P}*Z*E from having

a constant output, and large otherwise. To see that*η*b*Z* is positive, use the Schrödinger picture and let the

output ofC∗be the state*σ*for all inputs. Then note that infC*δ*(C,P*Z*E) =minCmax*zδ*(*σ,*E∗(*θz*)), where the

latter*δ*is the trace distance. Choosing*σ*= 1*d*

P

*z*E∗(*θz*)and using joint convexity of the trace distance, we

have infC*δ*(C,P*Z*E)≤ *d*−*d*1.

Z P*Z* _{E} *B* ≈*d*−*d*1−*η*b*Z* Z C *B*

Y Y

Figure 5: Figure of “demerit” version of preparation disturbance. Another approach to defining
preparation disturbance is to consider distinguishability to a non-ideal device instead of an ideal
device. The apparatusE imparts a large disturbance to the preparationP*Z*if the output of the

com-binationP*Z*E is essentially independent of the input. Thus we consider the distinguishability ofP*Z*E

and a constant preparation_{C} which outputs a fixed state regardless of the inputZ.

For finite-dimensional systems, all the measures of error and disturbance can be expressed as semidefinite
programs, as detailed in AppendixB. As an example, we compute these measures for the simple case of a
non-ideal*X*measurement on a qubit; we will meet this example later in assessing the tightness of the uncertainty
relations and their connection to wave-particle duality relations in the Mach-Zehnder interferometer. Consider
the ideal measurement isometry (6), and suppose that the basis states_{|}*b _{x}*

_{〉}are replaced by two pure states

_{|}

*γx*〉

which have an overlap_{〈}*γ*0|*γ*1〉=sin*θ*. Without loss of generality, we can take|*γx*〉=cos*θ*2|*bx*〉+sin*θ*2|*bx*+1〉.
The optimal measurement_{Q}for distinguishing these two states is just projective measurement in the _{|}*bx*〉

basis, so let us consider the channel_{EMZ}_{=}_{WQ}. Then, as detailed in AppendixB, for*Z*canonically conjugate
to*X* we find

*"X*(EMZ) =12(1−cos*θ*) and (14)

*νZ*(EMZ) =*ηZ*(E) =*η*b*Z*(E) = 12(1−sin*θ*). (15)

In all of the figures of merit, the optimal recovery map_{R}is to do nothing, while in*η*b*Z*the optimal channelC

outputs the average of the two outputs of_{P}*Z*E.

**4** **Uncertainty relations in finite dimensions**
**4.1** **Complementarity measures**

Before turning to the uncertainty relations, we first present several measures of complementarity that will
appear therein. Indeed, we can use the above notions of disturbance to define several measures of
comple-mentarity that will later appear in our uncertainty relations. For instance, we can measure the
complemen-tarity of two observables just by using the measurement disturbance*ν. Specifically, treating*_{Q}*X* as the actual

measurement and_{Q}*Z* as the ideal measurement, we define*cM*(*X*,*Z*):=*νZ*(Q*X*). This quantity is equivalent

to*"Z*(Q*X*)since any recovery mapRX→Zin*"Z*can be used to defineR0X→*A*in*νZ*byR0=RP*Z*. Similarly, we

could treat one observable as defining the ideal state preparation device and the other as the measurement
apparatus, which leads to*c _{P}*

_{(}

*X*,

*Z*

_{)}:=

*ηZ*(Q

*X*). Here we could also use the “figure of demerit” and define

b

*c _{P}*

_{(}

*X*,

*Z*

_{)}:=

*η*b

*Z*(Q

*X*).

Though the three complementarity measures are conceptually straightforward, it is also desireable to have
closed-form expressions, particularly for the bounds in the uncertainty relations. To this end, we derive lower
bounds as follows. First, consider*c _{M}* and choose as inputs

*Z*basis states. This gives, for random choice of input,

*cM*(*X*,*Z*)≥inf

R *δ*(P*Z*Q*Z*,P*Z*Q*X*R) (16a)

≥1_{−}max

*R*

1

*d*

X

*xz* |〈

*ϕx*|*θz*〉|2*Rz x* (16b)

≥1_{−}max

*R*

1

*d*

X

*x*

max

*z* |〈*ϕx*|*θz*〉|

2X

*z*0

*Rz*0*x* (16c)

=1−1*d*

X

*x*

max

*z* |〈*ϕx*|*θz*〉|

2_{,} _{(16d)}

where the maximization is over stochastic matrices*R*, and we use the fact thatP* _{z}Rz x* =1 for all

*x*. For

*cP*we

map_{R}_{X}_{→}_{Z}, we have

*cP*(*X*,*Z*)≥_{R}inf

X→*Aδ*(P*Z*Q*Z*

,_{P}*Z*Q*X*RQ*Z*) (17a)

=_{R}inf

X→Z

*δ*(P*Z*Q*Z*,P*Z*Q*X*R) (17b)

≥1_{−} 1

*d*

X

*x*

max

*z* |〈*ϕx*|*θz*〉|

2_{.} _{(17c)}

Forb*c _{P}*

_{(}

*X*,

*Z*

_{)}we have

b*c _{P}*

_{(}

*X*,

*Z*

_{) =}

*d*−1

*d* −_{C}_{:const.}inf *δ*(C,P*Z*Q*X*) (18a)

=*d*−*d*1−min* _{P}* max

_{z}*δ*(

*P*,Q∗

*X*(

*θz*)) (18b)

≥*d*−1

*d* −max* _{z}* 12
X

*x*

|1*d*− |〈*ϕx*|*θz*〉|2|, (18c)

where the bound comes from choosing *P* to be the uniform distribution. We could also choose *P*_{(}*x*_{) =}

|〈*ϕx*|*θz*0〉|2for some *z*0 to obtain the boundb*c _{P}*(

*X*,

*Z*)≥

*d*−1

*−min*

_{d}*0max*

_{z}*1*

_{z}_{2}P

*x*

Tr_{[}*ϕx*(*θz*−*θz*0)]

. However, from numerical investigation of random bases, it appears that this bound is rarely better than the previous one.

Let us comment on the properties of the complementarity measures and their bounds in (16d), (17c),
and (18c). Both expressions in the bounds are, properly, functions only of the two orthonormal bases
in-volved, depending only on the set of overlaps. In particular, both are invariant under relabelling the bases.
Uncertainty relations formulated in terms of conditional entropy typically only involve the largest overlap
or largest two overlaps[7,57], but the bounds derived here are yet more sensitive to the structure of the
overlaps. Interestingly, the quantity in (16d) appears in the information exclusion relation of[57], where
the sum of mutual informations different systems can have about the observables*X* and*Z* is bounded by
log2*d*P*x*max*z*|〈*ϕx*|*θz*〉|2.

The complementarity measures themselves all take the same value in two extreme cases: zero in the trivial
case of identical bases,_{(}*d*_{−}1_{)}*/d*in the case that the two bases are conjugate, meaning|〈*ϕx*|*θz*〉|2=1/*d*for

all *x*,*z*. In between, however, the separation between the two can be quite large. Consider two observables
that share two eigenvectors while the remainder are conjugate. The bounds (16d) and (17c) imply that*c _{M}*
and

*cP*are both greater than(

*d*−3)

*/d*. The bound onb

*cP*from (18c) is zero, though a better choice of constant

channel can easily be found in this case. In dimensions*d*_{=}3*k*_{+}2, fix the constant channel to output the
distribution*P*with probability 1/3 of being either of the last two outputs, 1/3*k*for any*k*of the remainder,
and zero otherwise. Then we have ˆ*cP*≥ *d*−1*d* −max*zδ*(*P*,Q∗*X*P*Z*∗(*z*)). It is easy to show the optimal value is 2/3

so that ˆ*cP*≥(*d*−3)*/3d*. Hence, in the limit of large*d*, the gap between the two measures can be at least 2/3.

This example also shows that the gap between the complementary measures and the bounds can be large, though we will not investigate this further here.

**4.2** **Results**

We finally have all the pieces necessary to formally state our uncertainty relations. The first relates measure-ment error and measuremeasure-ment disturbance, where we have

**Theorem 1.** *For any two observables X and Z and any quantum instrument*_{E}*,*
Æ

2"*X*(E) +*νZ*(E)≥*cM*(*X*,*Z*) *and* (19)

*"X*(E) +

Æ

2ν_{Z}_{(}_{E}_{)}_{≥}*c _{M}*

_{(}

*Z*,

*X*

_{)}. (20)

Due to Lemma1, any joint measurement of two observables can be decomposed into a sequential
measure-ment, which implies that these bounds hold for joint measurement devices as well. Indeed, we will make
use of that lemma to derive (20) from (19) in the proof below. Of course we can replace the*cM* quantities

with closed-form expressions using the bound in (16d). Figure6shows the bound for the case of conjugate
observables of a qubit, for which*cM*(*X*,*Z*) =*cM*(*Z*,*X*) =12. It also shows the particular relation between error
and measurement disturbance achieved by the apparatus_{EMZ}mentioned at the end of §3, from which we can
conclude the that bound is tight in the region of vanishing error or vanishing disturbance.

0 1_{/}_{2}
1_{/2}

Error

Disturbance

(*"X*,*νZ*)

(*"X*,*ηZ*)&(*"X*,*η*b*Z*)

EMZ

Figure 6: Error versus disturbance bounds for conjugate qubit observables. Theorem1restricts the
possible combinations of measurement error*"X* and measurement disturbance*νZ* to the dark gray

region bounded by the solid line. Theorem2additionally includes the light gray region. Also shown
are the error and disturbance values achieved by_{EMZ}from §3.

**Theorem 2.** *For any two observables X and Z and any quantum instrument*E*,*
Æ

2"*X*(E) +*ηZ*(E)≥*cP*(*X*,*Z*) *and* (21)

Æ

2"*X*(E) +*η*b*Z*(E)≥b*cP*(*X*,*Z*). (22)

Returning to Figure6but replacing the vertical axis with*ηZ*or*η*b*Z*, we now have only the upper branch of the

bound, which continues to the horizontal axis as the dotted line. Here we can only conclude that the bounds are tight in the region of vanishing error.

**4.3** **Proofs**

The proofs of all three uncertainty relations are just judicious applications of the triangle inequality, and
the particular bound comes from the setting in which_{P}*Z* meetsQ*X*. We shall make use of the fact that an

instrument which has a small error in measuring_{Q}*X* is close to one which actually employs the instrument

associated with_{Q}*X*. This is encapsulated in the following

**Lemma 2.** *For any apparatus*_{E}*A*→Y*B* *there exists a channel* FX*A*→Y*B* *such that* *δ*(E,Q0*X*F)≤

p
2"*X*(E)*,*

*where*_{Q}0

*X* *is a quantum instrument associated with the measurement*Q*X. Furthermore, if*Q*Xis a projective*

*measurement, then there exists a state preparation*_{P}X→Y*Bsuch thatδ*(E,Q*X*P)≤p2"*X*(E)*.*

*Proof.* Let*V* :_{H}*A*→H*B*⊗H*E*⊗*L*2(X)and*WX*:H*A*→*L*2(X)⊗H*A*be respective dilations ofE andQ*X*. Using

the dilation*WX* we can define the instrumentQ0*X* as

Q0*X*:*L*∞(X)⊗B(H*B*)→B(H*A*)

*g*_{⊗}*A*_{7→}*W*∗

*X*(*π*(*g*)⊗*A*)*WX*.

(23)

Suppose_{R}_{Y}_{→}_{X}is the optimal map in the definition of*"X*(E), and letR0Y→XY be the extension ofRwhich
keeps the inputY; it has a dilation*V*0:*L*2_{(}Y_{)}_{→}*L*2_{(}Y_{)}_{⊗}*L*2_{(}X_{)}. By Stinespring continuity, in finite dimensions

there exists a conditional isometry*U _{X}*:

*L*2

_{(}

_{X}

_{)}

_{⊗}

_{H}

*A*→*L*2(X)⊗*L*2(Y)⊗H*B*⊗H*E* such that

*V*0_{V}_{−}_{U}

*XWX*

∞≤ Æ

2"*X*(E). (24)

Now consider the map

E0:*L*∞(Y_{)}⊗_{B}_{(}_{H}*B*)→B(H*A*)

*f* _{⊗}*A*_{7→}*W*∗

*XUX*∗(1X⊗*π*(*f*)⊗*A*⊗1*E*)*UXWX*.

By the other bound in Stinespring continuity we thus have*δ*(E,E0)≤p2"*X*(E). Furthermore, as described

in §2.4, *UX* is a conditional isometry, i.e. a collection of isometries*Ux* :H*A* → *L*2(Y)⊗H*B*⊗H*E* for each

measurement outcome*x*. Note that we may regard elements of *L*∞_{(}_{X}_{)}_{⊗}_{B}_{(}_{H}_{)}_{as sequences}_{(}_{A}

*x*)*x*∈Xwith
*Ax*∈B(H)for all *x*∈Xsuch that ess sup*x*k*Ax*k∞*<*∞. Therefore we may define

F:*L*∞(Y_{)}_{⊗}_{B}_{(}_{H}_{B}_{)}_{→}*L*∞_{(}X_{)}_{⊗}_{B}_{(}_{H}_{A}_{)}

*f* ⊗*A*7→(*Ux*∗(*π*(*f*)⊗*A*⊗1*E*)*Ux*)*x*∈X,

(26)

so that_{E}0_{=}_{Q}0

*X*F. This completes the proof of the first statement.

If _{Q}*X* is a projective measurement, then the output*B* of Q0*X* can just as well be prepared from the X

output. Describing this with the map_{P}0

X→X*A*which prepares states in*A*given the value ofXand retainsXat

the output, we have_{Q}0

*X*=Q*X*P0. SettingP=P0Fcompletes the proof of the second statement.

Now, to prove (19), start with the triangle inequality and monotonicity. Suppose PX→Y*B* is the state

preparation map from Lemma2. Then, for any_{R}_{Y}_{B}_{→}* _{A}*,

*δ*(Q*Z*,Q*X*PRQ*Z*)≤*δ*(Q*Z*,ERQ*Z*) +*δ*(ERQ*Z*,Q*X*PRQ*Z*) (27a)

≤*δ*(Q*Z*,ERQ*Z*) +*δ*(E,Q*X*P) (27b)

=*δ*(Q*Z*,ERQ*Z*) +

Æ

2"*X*(E). (27c)

Observe that_{PRQ}* _{Z}* is just a map

_{R}0

X→Z. Taking the infimum overRwe then have Æ

2"*X*(E) +*νZ*(E)≥inf_{R} *δ*(Q*Z*,Q*X*PRQ*Z*) (28a)

≥inf

R *δ*(Q*Z*,Q*X*R). (28b)

To show (20), let_{R}_{Y}_{B}_{→}* _{A}*and

_{R}0

Y→X be the optimal maps in*νZ*(E)and*"X*(E), respectively. Now apply
Lemma1to_{M}_{=}_{ER}0_{RQ}

*Z*and suppose thatE*A*0→Z*B*is the resulting instrument andMZ*B*→Xis the conditional
measurement. By the above argument,p2"_{Z}_{(}_{E}0_{) +}_{ν}_{X}_{(}_{E}0_{)}_{≥}_{inf}

R*δ*(Q*X*,Q*Z*R). But*"Z*(E0)≤*δ*(Q*Z*,E0T*B*) =

*νZ*(E)and*νX*(E0)≤*δ*(Q*X*,E0M) =*"X*(E), where in the latter we use the fact that we could always reprepare

an*X* eigenstate and then let_{Q}*X*measure it. Therefore the desired bound holds.

To establish (21), we proceed just as above to obtain

*δ*(P*Z*,P*Z*Q*X*PR)≤*δ*(P*Z*,P*Z*ER) +

Æ

2"_{X}_{(}_{E}_{)}. (29)

Now_{P}X→Y*B*RY*B*→*A*is a preparation mapPX→*A*, and taking the infimum overRgives

Æ

2"*X*(E) +*ηZ*(E)≥inf

R *δ*(P*Z*,P*Z*Q*X*PR) (30a)

≥inf

P *δ*(P*Z*,P*Z*Q*X*P). (30b)

Finally, (22). Since the*η*_{b}*Z* disturbance measure is defined “backwards”, we start the triangle inequality

with the distinguishability quantity related to disturbance, rather than the eventual constant of the bound.
For any channel_{C}_{Z}_{→}_{X}and_{P}_{X}_{→}_{Y}*B*from Lemma2, just as before we have

*δ*_{(}_{CP},_{P}*Z*E)≤*δ*(CP,P*Z*Q*X*P) +*δ*(P*Z*Q*X*P,P*Z*E) (31a)

≤*δ*(C,P*Z*Q*X*) +

Æ

2"*X*(E). (31b)

Now we take the infimum over constant channels_{C}Z→X. Note that

inf

CZ→Y*Bδ*(C

,_{P}*Z*E)≤ inf
CZ→X

*δ*(CP,P*Z*E). (32)

Therefore, we have

Æ

2"*X*(E) +*η*b*Z*(E)≥ *d*−1*d* −inf_{C} *δ*(C,P*Z*Q*X*). (33)

This last proof also applies to a more general definition of disturbance which does not use_{P}*Z*at the input,

be thought of as the result of performing an ideal*Z* measurement, but forgetting the result. More formally,
letting_{Q}*\*

*Z*=W*Z*TZwithW*Z*:*a*→*WZ*∗*aWZ*, we can define

e

*ηZ*(E) = *d*−1*d* −inf_{C} *δ*(C,Q*\Z*E). (34)

Though perhaps less conceptually appealing, this is a more general notion of disturbance, since now we can
potentially use entanglement at the input to increase distinguishability of_{Q}*\*

*Z*E from any constant channel.

However, due to the form of_{Q}*\*

*Z*, entanglement will not help. Applied to any bipartite state, the mapQ*\Z*

produces a state of the formP* _{z}pz*|

*θz*〉〈

*θz*| ⊗

*σz*for some probability distribution

*pz*and set of normalized

states*σz*, and therefore the input toEitself is again an output ofP*Z*. Since classical correlation with ancillary

systems is already covered in*η*b*Z*(E), it follows that*η*e*Z*(E) =*η*b*Z*(E).

**5** **Position & momentum**

**5.1** **Gaussian precision-limited measurement and preparation**

Now we turn to the infinite-dimensional case of position and momentum measurements. Let us focus on
Gaussian limits on precision, where the convolution function*α* described in §2.2 is the square root of a
normalized Gaussian of width*σ, and for convenience define*

*g _{σ(}x*

_{) =}

_{p}1 2πσ

*e*

−_{2}*x _{σ}*22 . (35)

One advantage of the Gaussian choice is that the Stinespring dilation of the ideal*σ-limited measurement*
device is just a canonical transformation. Thus, measurement of position*Q*just amounts to adding this value
to an ancillary system which is prepared in a zero-mean Gaussian state with position standard deviation*σQ*,

and similarly for momentum. The same interpretation is available for precision-limited state preparation.
To prepare a momentum state of width *σP*, we begin with a system in a zero-mean Gaussian state with

momentum standard deviation*σP* and simply shift the momentum by the desired amount.

Given the ideal devices, the definitions of error and disturbance are those of §3, as in the finite-dimensional
case, with the slight change that the first term of*η*_{b}is now 1. To reduce clutter, we do not indicate*σQ*and*σP*

specifically in the error and disturbance functions themselves.

Since our error and disturbance measures are based on possible state preparations and measurements
in order to best distinguish the two devices, in principle one ought to consider precision limits in the
distin-guishability quantity*δ. However, we will not follow this approach here, and instead we allow test of arbitrary*
precision in order to preserve the link between distinguishability and the cb norm. This leads to bounds that
are perhaps overly pessimistic, but nevertheless limit the possible performance of any device.

**5.2** **Results**

As discussed previously, the disturbance measure of demerit*η*_{b}cannot be expected to lead to uncertainty
relations for position and momentum observables, as any non-constant channel can be perfectly differentiated
from a constant one by inputting states of arbitrarily high momentum. We thus focus on the disturbance
measures of merit.

**Theorem 3.** *Set c*_{=}2σ*QσP* *for any precision valuesσQ*,*σP* *>*0*. Then for any quantum instrument*E*,*

Æ

2"*Q*(E) +*νP*(E)

*"Q*(E) +Æ2ν*Q*(E)

)

≥ 1−*c*2

(1+*c*2*/*3_{+}* _{c}*4

*/*3

_{)}3

*/*2

*and*(36)

q

2"*Q*(E) +*ηP*(E)≥ (1+*c*

2_{)}1*/*2

((1+*c*2) +*c*2*/*3_{(}_{1}_{+}* _{c}*2

_{)}2

*/*3

_{+}

*4*

_{c}*/*3

_{(}

_{1}

_{+}

*2*

_{c}_{)}1

*/*3

_{)}3

*/*2. (37)

Before proceeding to the proofs, let us comment on the properties of the two bounds. As can be seen in
Figure7, the bounds take essentially the same values for*σQσP* 12, and indeed both evaluate to unity at
*σQσP* =0. This is the region of combined position and momentum precision far smaller than the natural

scale set byħ*h*, and the limit of infinite precision accords with the finite-dimensional bounds for conjugate

The distinction between these two cases is a result of allowing arbitrarily precise measurements in the
dis-tinguishability measure. It can be understood by the following heuristic argument. Consider an experiment
in which a momentum state of width*σ*in* _{P}* is subjected to a position measurement of resolution

*σQ*and then a

momentum measurement of resolution*σ*out* _{P}* . From the uncertainty principle, we expect the position
measure-ment to change the momeasure-mentum by an amount∼1/σ

*Q*. Thus, to reliably detect the change in momentum,

*σ*out* _{P}* must fulfill the condition

*σ*out

_{P}*σ*in

*+1/σ*

_{P}*Q*. The Heisenberg limit in the measurement disturbance

scenario is*σ*out* _{P}* =2/σ

*Q*, meaning this condition cannot be met no matter how small we choose

*σ*in

*P*. This is

consistent with no nontrivial bound in (36) in this region. On the other hand, for preparation disturbance the
Heisenberg limit is*σ*in* _{P}* =2/σ

*Q*, so detecting the change in momentum simply requires

*σ*out

*P*1/σ

*Q*. A more

satisfying approach would be to include the precision limitation in the distinguishability measure to restore the symmetry of the two scenarios, but this requires significant changes to the proof and is left for future work.

1* _{/2}* 1

1

0

*σQσP*

lower

bound

measurement preparation

Figure 7: Uncertainty bounds appearing in Theorem3in terms of the combined precision*σQσP*.

The solid line corresponds to the bound involving measurement disturbance, (36), the dashed line to the bound involving preparation disturbance, (37).

**5.3** **Proofs**

The proof of Theorem3is broadly similar to the finite-dimensional case. We would again like to begin with
FQ*A*→Y*B*from Lemma2such that*δ*(E,Q0*Q*F)≤

Æ

2"*Q*(E). However, the argument does not quite go through,

as in infinite dimensions we cannot immediately ensure that the infimum in Stinespring continuity is attained.
Nonetheless, we can consider a sequence of maps_{(}_{F}*n*)*n*∈Nsuch that the desired distinguishability bound holds

in the limit*n*_{→ ∞}.

To show (36), we follow the steps in (27). Now, though, consider the map_{F}0

*n*which just appendsQto the

output ofF*n*, and defineN=Q0*Q*F*n*RQ*P*, whereQ*Q*0 is the instrument associated with position measurement

Q*Q*. Then we have

*δ*(Q*P*,N TQ)≤*δ*(Q*P*,ERQ*P*) +*δ*(ERQ*P*,N TQ) (38a)

≤*δ*(Q*P*,ERQ*P*) +*δ*(E,Q0*Q*F*n*). (38b)

Taking the limit*n*_{→ ∞}and the infimum over recovery maps_{R}producesÆ2"_{Q}_{(}_{E}_{) +}*νP*(E)on the righthand

side. We can bound the lefthand side by testing with pure unentangled inputs:

*δ*(Q*P*,N TQ)≥sup

*ψ*,*f* 〈*ψ,* Q*P*(*f*)−[N TQ](*f*)

*ψ*〉. (39)

Now we want to show that, since_{Q}* _{P}* is covariant with respect to phase space translations, without loss
of generality we can take

_{N}to be covariant as well. Consider the translated version of both

_{Q}

*and*

_{P}_{N T}

_{Q}, obtained by shifting their inputs and outputs correspondingly by some amount

*z*

_{= (}

*q*,

*p*

_{)}. For the states

*ψ*this shift is implemented by the Weyl-Heisenberg operators

*Vz*, while for tests

*f*only the value of

*p*is

to the work of Werner[22]for furter details. Since_{T}_{Q}just ignores theQoutput of the measurement_{N}, we

may thus proceed by assuming that_{N} is a covariant measurement.
Any covariant_{N} has the form

N(*f*) =
Z

R2

d*z*

2π *f*(*z*)*VzmVz*∗, (40)

for some positive operator*m*such that Tr_{[}*m*_{] =}1. Due to the definition of _{N}, the position measurement
result is precisely that obtained from_{Q}*Q*. By the covariant form ofN, this implies that the position width of

*m*is just*σQ* (or rather that of the parity version of*m*, see[22]). Suppose the momentum distribution has

standard deviation*σ*b*P*; then*σQσ*b*P*≥1/2 follows from the Kennard uncertainty relation[3].

Now we can evaluate the lower bound term by term. Let us choose a Gaussian state in the momentum
representation and test function: *ψ* = *g*12

*σψ* and *f* =

p_{2πσ}

*fgσf*. Then the first term is a straightforward
Gaussian integral, since the precision-limited measurement just amounts to the ideal measurement convolved
with*gσP*:

〈*ψ,*Q*P*(*f*)*ψ*〉=

Z

R2

*d p*0_{d p g}*σψ*(*p*

0_{)}_{g}*σP*(*p*

0_{−}_{p}_{)}_{f}_{(}_{p}_{)} _{(41a)}

=Ç *σf*

*σ*2* _{f}*+

*σ*2

*+*

_{P}*σ*2

_{ψ}. (41b)

The second term is the same, just with*σ*b*P* instead of*σP*, so we have

*δ*(Q*P*,N TQ)≥

*σf*

Ç

*σ*2* _{f}* +

*σ*2

*+*

_{P}*σ*2

*−*

_{ψ}*σf*

Ç

*σ*2* _{f}*+

*σ*b2

*+*

_{P}*σ*2

_{ψ}. (42)

The tightest possible bound comes from the smallest*σ*b*P*, which is 1/2σ*Q*, and the bound is clearly trivial if

*σQσP* ≥1/2. If this is not the case, we can optimize our choice of*σf*. To simplify the calculation, assume

that*σψ* is small compared to*σf* (so that we are testing with a very narrow momentum state). Then, with

*c*_{=}2σ* _{Q}σP*, the optimal

*σf*is given by

*σ*2* _{f}* =

*σ*2

*P*

*c*2*/*3_{(}_{1}_{+}* _{c}*2

*/*3

_{)}. (43) Using this in (42) gives (36).

For preparation disturbance, proceed as before to obtain

*δ*(P*P*,P*P*Q0*Q*F*n*0RTQ)≤*δ*(P*P*,P*P*ER) +*δ*(P*P*ER,P*P*Q*Q*0F*n*0RTQ) (44a)

≤*δ*(P*P*,P*P*ER) +*δ*(E,Q0*Q*F*n*) (44b)

Now the limit*n*_{→ ∞}and the infimum over recovery maps_{R}producesÆ2"*Q*(E) +*ηP*(E)on the righthand

side. A lower bound on the quantity on the lefthand side can be obtained by usingP*P* to prepare a*σP*-limited

input state and making a*σm*-limited momentum measurement ¯Q*P* measurement on the output, so that, for

N as before,

*δ*(P*P*,P*P*Q0*Q*F*n*0RTQ)≥ sup

*ψ*:Gaussian;*f*〈*ψ, ¯*Q*P*(*f*)−[N TQ](*f*)

*ψ*〉. (45)

The only difference to (39) is that the supremum is restricted to Gaussian states of width*σP*. The covariance

argument nonetheless goes through as before, and we can proceed to evaluate the lower bound as above. This yields

*δ*(P*P*,P*P*Q0*Q*F*n*0RTQ)≥

*σf*

Ç

*σ*2* _{f}*+

*σ*2

*+*

_{m}*σ*2

*−*

_{P}*σf*

r

*σ*2* _{f}* +4

*σ*12

*+*

_{Q}*σ*2

*P*

. (46)

We may as well consider*σm*→0 so as to increase the first term. The optimal*σf* is then given by the optimizer

**6** **Applications**

**6.1** **No information about***Z***without disturbance to***X*

A useful tool in the construction of quantum information processing protocols is the link between reliable
transmission of*X*eigenstates through a channelNand*Z*eigenstates through its complementN*]*_{, particularly}
when the observables *X* and *Z* are maximally complementary, i.e._{|〈}*ϕx*|*ϑz*〉|2 = 1*d* for all *x*,*z*. Due to the

uncertainty principle, we expect that a channel cannot reliably transmit the bases to different outputs, since
this would provide a means to simultaneously measure*X* and*Z*. This link has been used by Shor and Preskill
to prove the security of quantum key distribution[58]and by Devetak to determine the quantum channel
capacity[59]. Entropic state-preparation uncertainty relations from[6,44]can be used to understand both
results, as shown in[60,61].

However, the above approach has the serious drawback that it can only be used in cases where the specific
*X*-basis transmission over_{N} and*Z*-basis transmission over_{N}*]*_{are in some sense compatible and not}_{}*coun-terfactual*; because the argument relies on a state-dependent uncertainty principle, both scenarios must be
compatible with the same quantum state. Fortunately, this can be done for both QKD security and quantum
capacity, because at issue is whether*X*-basis (*Z*-basis) transmission is reliable (unreliable) on average*when*
*the states are selected uniformly at random*. Choosing among either basis states at random is compatible with
a random measurement in either basis of half of a maximally entangled state, and so both*X* and*Z*basis
sce-narios are indeed compatible. The same restriction to choosing input states uniformly appears in the recent
result of[33], as it also ultimately relies on a state-preparation uncertainty relation.

Using Theorem 2we can extend the method above to counterfactual uses of arbitrary channels_{N}, in
the following sense: If acting with the channel_{N} does not substantially affect the possibility of performing
an*X* measurement, then*Z*-basis inputs to_{N}*]*_{result in an essentially constant output. More concretely, we}
have

**Corollary 1.** *Given a channel*N *and complementary channel*N*] _{, suppose that there exists a measurement}*

*ΛX*

*such thatδ*(Q

*X*,N

*ΛX*)≤

*". Then there exists a constant channel*C

*such that*

*δ*(Q*\Z*N*]*,C)≤

p

2"_{+} *d*−1

*d* −b*cP*(*X*,*Z*). (47)

*For maximally complementary X and Z,δ*(Q*\Z*N*]*,C)≤

p_{2"}_{.}

*Proof.* Let*V* be the Stinespring dilation of_{N} such that_{N}*]* _{is the complementary channel and define}

E =

VN*ΛX*. For C the optimal choice in the definition of *η*b*Z*(E), (22), (34), and *η*e*Z* = *η*b*Z* imply *δ*(Q*\Z*E,C) ≤

p_{2"}

+*d*−*d*1−b*cP*(*X*,*Z*). SinceN*]*is obtained fromE by ignoring the*ΛX* measurement result,*δ*(Q*\Z*N*]*,C)≤

*δ*(Q*\Z*E,C).

This formulation is important because in more general cryptographic and communication scenarios we are
interested in the worst-case behavior of the protocol, not the average case under some particular probability
distribution. For instance, in [46]the goal is to construct a classical computer resilient to leakage of *Z*
-basis information by establishing that reliable*X* basis measurement is possible despite the interference of the
eavesdropper. However, such an*X* measurement is entirely counterfactual and cannot be reconciled with the
actual*Z*-basis usage, as the*Z*-basis states will be chosen*deterministically*in the classical computer.

It is important to point out that, unfortunately, calibration testing is in general completely insufficient
to establish a small value of*δ*(Q*X*,N*ΛX*). More specifically, the following example shows that there is no

dimension-independent bound connecting inf*ΛXδ*(Q*X*,N*ΛX*)to the worst case probability of incorrectly
iden-tifying an *X* eigenstate input to _{N}, for arbitrary _{N}. Let the quantities *p _{yz}* be given by

*p*

_{y}_{,0}

_{=}2/

*d*for

*y*

_{=}0, . . . ,

*d/2*−1,

*p*

_{y}_{,1}

_{=}2/

*d*for

*y*

_{=}

*d/2, . . . ,d*

_{−}1, and

*p*

_{y}_{,}

_{z}_{=}1/

*d*otherwise, where we assume

*d*is even, and then define the isometry

*V*:H

*A*→H

*B*⊗H

*C*⊗H

*D*as the map taking|

*z*〉

*A*toP

*y*p

*pyz*|

*y*〉

*B*|

*z*〉

*C*|

*y*〉

*D*.

Finally, let_{N}:_{B}_{(}_{H}_{B}_{)}_{⊗}_{B}_{(}_{H}_{C}_{)}_{→}_{B}_{(}_{H}_{A}_{)}be the channel obtained by ignoring*D*, i.e. in the Schrödinger picture
N∗(*%*) =Tr*D*[*V%V*∗]. Now consider inputs in the*X* basis, with*X* canonically conjugate to*Z*. As shown in

AppendixC, the probability of correctly determining any particular*X* input is the same for all values, and
is equal to * _{d}*12

P

*y*

P

*z*p*py*,*z*

2

= (*d*+p2−2_{)}2* _{/}_{d}*2

_{. The worst case}

_{X}_{error probability therefore tends to}zero like 1/

*d*as

*d*

_{→ ∞}. On the other hand,

*Z*-basis inputs 0 and 1 to the complementary channelE

*]*

_{result}in completely disjoint output states due to the form of

*p*. Thus, if we consider a test which inputs one of these randomly and checks for agreement at the output, we find infC

_{yz}*δ*(Q

*\*